Martinez-Vazquez, P, Gkantou, M and Baniotopoulos, C

Strength and ductility demands on wind turbine towers due to earthquake and wind load

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Martinez-Vazquez, P, Gkantou, M and Baniotopoulos, C (2018) Strength and ductility demands on wind turbine towers due to earthquake and wind load. Proceedings of the Institution of Civil Engineers - Structures and Buildings. pp. 1-8. ISSN 0965-0911

LJMU Research Online

Strength and ductility demands on wind turbine towers due to earthquake

and wind load

P. Martinez-Vazquez, M. Gkantou, C. Baniotopoulos

School of Engineering, Civil Engineering Department, The University of Birmingham, Edgbaston B15 2TT,

Birmingham, United Kingdom

Abstract

In earthquake prone areas, wind and earthquake loads are assumed to be statistically uncorrelated, therefore their

interaction is ignored by existing design guidelines. However, the fact that strong earthquake events are commonly

followed by aftershocks and that wind is constantly flowing at high speeds around wind farms increase the

probability of their joint occurrence, thus making current design assumptions questionable. This investigation

shows that multi-hazard scenarios magnify strength demands of wind turbine towers designed against isolated

load conditions, hence modifying their performance level. It is also shown that, under certain conditions, the

probabilities associated to the joint occurrence of earthquake and low to strong wind events match or exceed those

related to the original design, thus rendering wind energy infrastructure susceptible to unforeseen damage.

Key words: Earthquake; Wind; Load Interaction; Multi-hazard; Performance level; Wind turbine

tower

Notation

A Area exposed to wind Γ Characteristic structural density

CD Drag coefficient β Modal exponential

Cprob Probability factor ϒ Generalised quantity

D Section diameter φ Modal shape

F Force μ Ductility factor

H Height of building ρ Air density

Iu Turbulent intensity σ Standard deviation of wind velocity

Ms Magnitude scale ω Natural frequency

N Number of earthquake events ξ Structural damping

PJ Probability associated to load J

PGA Peak ground acceleration

Rμ Strength reduction factor

Ū Average wind velocity

Vs30 Upper 30-m mean shear wave velocity

a, b Site-dependent parameters

d Structural displacement

m Structural mass

n0 Fundamental frequency of vibration

p Load

s Load gradient

t Section thickness

uy Yield displacement

z Vertical coordinate

Martinez-Vazquez, P., Gkantou, M. and Baniotopoulos, C. (2018) Strength and ductility demands on

wind turbine towers due to earthquake and wind load. Proceedings of the Institution of Civil Engineers–

Structures and Buildings. https://doi.org/10.1680/jstbu.17.00154

1. Introduction

In the past decades, regions across the world have been identified where the energy production of Class

I and Class II wind energy increases exponentially. This is largely due to the availability of natural

resources, technological developments and qualified labour. The fast growth of energy generation

however requires careful consideration of the type of infrastructure that can fulfil the demands in terms

of strength, resilience and innovation. In earthquake-prone regions, tectonic and environmental

conditions make infrastructure often susceptible to earthquakes and wind effects both during

construction and once in operation. Notwithstanding that, current engineering practice disregards their

simultaneous occurrence even though past earthquake records show that further ground accelerations

can occur within days or even hours from main events. Examples of this include the earthquake that hit

the Sichuan Province in China in 2008 (Ms = 7.9) which was followed by 12 weeks with 42 aftershocks

ranging in magnitude between 5 < Ms < 6.4, killing over 87,000 and leaving over £56bn in losses

(Daniell et al., 2012). The Great East Japan Earthquake felt in 2011 (Ms = 9.0), killing more than 15,000,

was followed by 408 aftershocks (Ms > 5.0) within 96 hr, 68 of those of Ms > 6.0 and 5 of Ms > 7.0

(Matsutani, 2011). More recently, the earthquakes that hit Ecuador in 2016 (Ms = 7.8) killing over 600,

were followed by over 55 aftershocks in the first 24 hours (Shankar, 2016).

Past research has contributed to better understand the mechanisms through which extreme load events

affect infrastructure. Martinez-Vazquez (2017) showed that the combination of earthquakes and wind

would decrease the value of strength reduction factors that are calculated by ignoring the impact of

wind during an earthquake. Kiyomiya et al. (2002) suggested that wind turbines have adequate

earthquake resistance provided these are designed against typhoons, which could be the case of offshore

wind turbines but is not shared practice for onshore infrastructure design. Furthermore, Diaz and Suarez

(2014) demonstrated that an operational earthquake combined with design wind load tend to over-stress

the tower section hence increasing the strength demand established under isolated wind conditions. The

present paper thus aims at enhancing our understanding of multi-hazard load conditions and their effect

on wind energy infrastructure.

The paper is organised as follows, Section 2 describes the assembling process of two databases, one

containing historical earthquake records and the other containing simulated wind fields. Section 3

focuses on the estimation of generalised forces acting on three wind turbine towers and the calculation

of their dynamic response. The associated probabilities are discussed in Section 4 whilst the estimation

of strength and ductility demands related to multi-hazard scenarios is presented in Section 5. Some final

remarks are provided in Section 6.

2. Earthquake and wind record database

Earthquake records of magnitude 5.3 < Ms < 7.36 recorded on alluvium and with distances from

geological faults of up to 57 km were downloaded from the PEER (2016) ground motion database.

These records are assumed to be representative of alluvium and firm soils. It has been shown in Miranda

(1993) that without much variation they would produce similar strength demands to structures located

on either soil type. These records have a duration which oscillates between 30 and 80 s and were

measured at a time interval of 0.1 s. The list of historic earthquake records is provided in Table 1 which

also shows the associated epicentral distance, shear wave velocity (vs30), and peak ground acceleration

(PGA).

TABLE 1

On the other hand, a wind record database was established based on the simulation procedure reported

in Martinez-Vazquez and Rodriguez-Cuevas (2007) which follows the conditional simulation proposed

by Vanmarcke et al. (1993). The simulation algorithm requires knowledge of recorded data in at least

two points within the region of interest and enables inferring properly correlated wind data series at

intermediate points. The two initial data series were calculated by using classical Monte Carlo

techniques whereas intermediate points were calculated at 11 stations covering 250 m along a vertical

axis. The mean velocity �̅� and turbulence intensity I - defined as the ratio between the standard deviation

and the mean (σ/�̅�) - are shown for each simulation point in Table 2 for the case in which �̅� =20 ms-1.

This table also shows how simulated (s) and theoretical (t) values compare.

TABLE 2

According to the data in Table 2, the average ratio �̅�𝑡/�̅�𝑠 is 1.028 whereas the mean square error

associated to the simulated turbulence intensity is 0.002. The theoretical wind velocity and turbulence

intensity were determined based on a standard exponential law i.e. Uz = (z/zref)α using α = 0.22 and Iu,10

= 0.295 which corresponds to suburbs. Table 3 shows the target and calculated cross-correlation in the

lower and upper triangular matrices respectively, where Point 1 corresponds to that located at z = 10 m.

It can be seen in the table that the accuracy of the simulation increases with the proximity between

points – see for example the values around the main diagonal. The overall mean square error across

cross-correlation results is 0.0073 which was considered acceptable for this investigation.

TABLE 3

3. Generalised forces acting on wind turbines and related dynamic effects

Three wind turbine towers of 150 m, 200 m and 250 m height were identified. These are assumed to be

made of steel with specific weight of 7850 kgm-3, Young’s modulus of 200×109 Nm-2, damping level

of 5%, and having variable section across their length whilst fixed at their base. The geometry and

natural vibration frequency of each tower are shown in Table 4.

TABLE 4

Earthquake forces (FEQ) are proportional to the mass of the structure whilst wind forces (FW) were

derived from Bernoulli’s principle: FW=1⁄2 ρ CD A �̅�2 – where ρ is the density of the air, CD is a drag

coefficient (taken as 1.4), and A is the projected area of the segment exposed to wind. Generalised

forces, FEQ* and FW

*, were calculated by using Eq. (1), where ϒ (z) represents force or structural mass

per unit length, z is a vertical coordinate, λ equals 1 and 2 for F* and M* respectively, ϕ is the

fundamental modal shape which was approximated by ϕ (z) = (z/H)β – with β = 1.5 whilst taking H as

the height of the tower.

Υ∗ = ∫ 𝜙(𝑧)𝜆Υ(𝑧)𝑑𝑧𝐻

0 (1)

Generalised forces were combined to find the total amount acting on the wind turbines. This is shown

in Table 5 for seven levels of wind, including the case in which �̅� equals zero. This table also shows

the average estimated ratio FEQ*/ FW

*. Earthquake loading dominates for low values of wind velocity,

however this condition changes rapidly as the value of �̅� increases. For example when �̅� = 5 ms-1

earthquake are about twice wind forces whilst when �̅� = 10 ms-1 earthquake are about half wind forces.

At the point of the design wind speed, earthquake forces are about 12% of wind peak forces.

TABLE 5

Total generalised forces reported in Table 5 were used to determine the dynamic response of the wind

turbines listed in Table 4, assuming linear-elastic performance. This was done through the numerical

integration of Eq. (2) whose solution is given by Eq. (3).

𝑚�̈� + 𝑐�̇� + 𝑘𝑑 = 𝑝(𝑡) (2)

𝑑(𝑡) = 𝑒−𝜉𝜔Δ𝑡 [(𝑑(0) −𝑝𝑖

𝑘+

2𝑠𝜉

𝜔𝑛𝑘) cos(𝜔𝐷Δ𝑡) + (�̇�(0) + 𝑑(0)𝜉𝜔𝑛 −

𝑝𝑖𝜉𝜔𝑛

𝑘+

2𝑠𝜉2

𝑘−

𝑠

𝑘)

𝑠𝑖𝑛(𝜔𝐷Δ𝑡)

𝜔𝐷]

𝑝𝑖

𝑘+

𝑠Δ𝑡

𝑘−

2𝑠𝜉

𝜔𝑛𝑘

(3)

In Eq. (2) and (3) m, c, and k represent the system’s mass, damping coefficient, and stiffness; d(t) and

�̇�(𝑡) are displacement and velocity of response respectively; p(t) is the time-varying force; pi and s

represent the i-th force and its gradient, in the context of the numerical integration; ωn and ωD are the

undamped and damped frequency whilst ξ represents the fraction of critical damping. The results of the

numerical integration are shown in Table 6.

TABLE 6

The results in Table 6 show a considerable increase of peak displacements for relatively low levels of

wind acting on assumed earthquake-resisting structures. For example, when �̅� = 2.5 ms-1 the estimated

increase is of 21%, 35%, and 38% on the towers of 150 m, 200 m, and 250 m tall, respectively, whereas

when �̅� = 5 ms-1 those figures become 114%, 170%, and 219%. This suggests that the combined effect

of earthquake and wind load can be significantly higher than those due to earthquakes only, even for

relatively low levels of wind. However, infrastructure could also be designed to withstand wind load

only in which case the earthquake load would have a different level of impact depending on the design

wind load. Wind-resisting design is examined in more detail in the following sections.

4. Probabilities associated to wind and earthquake events

Eurocode 1 (European Standard, 2010) associates wind design loads to a 50-year return period whose

probability of annual exceedance is P50 = 0.02. The norm also provides Eq. (4) to scale the 10-min

design wind velocity, �̅�50 for other probability levels.

𝐶𝑝𝑟𝑜𝑏 = [1 − 𝐾 ∙ 𝑙𝑛(−𝑙𝑛(1 − 𝑝))

1 − 𝐾 ∙ 𝑙𝑛(−𝑙𝑛(0.98))]

𝑛

(4)

In Eq. (4) K is a shape parameter that depends on the coefficient of variation of the extreme-value

distribution, n is a constant number, Cprob is the probability factor and p represents probability. Eurocode

1 recommends K = 0.2 and n = 0.5.

𝑙𝑜𝑔10𝑁 = 𝑎 − 𝑏𝑀 (5)

On the other hand, the Gutenberg-Richter law quoted in Eq. (5), establishes a relationship between

frequency of earthquake events and their magnitude. In this equation, N represents the number of events

happening within a year, having magnitude > M and a, b are site-dependent constants. It follows that

Eq. (4) and (5) can be used to infer the magnitude of earthquake events associated to probability values

(PEQ) which, paired with those related to wind (PW), match the standard probability of wind-resisting

design i.e. PEQ ∙ PW = 0.02. This is shown in Table 7 for a set of randomly selected locations across the

world, assuming �̅�50 = 20 ms-1.

TABLE 7

In Table 7, rows 3 and 1 relate to local average wind speeds and their probability of occurrence as per

Eq. (4). Row 4 provides the probability associated to seismic magnitude depending on location. There

are five countries listed in Table 7 with their respective value of Ms which directly links to their

probability of occurrence as per Eq. (5). The reason those specific probabilities - quoted in rows 1 and

4 of the Table, have been selected is because their product matches the probability associated to the

design wind speed, which according to Eurocode 1 (European Standard, 2010), corresponds to the

design wind speed here assumed to be of 20 ms-1. Hence these are meaningful combinations derived

from the simultaneous occurrence of earthquake and wind events. According to the data shown in Table

7, �̅� = 10.33 ms-1 would always be exceeded in a year – see rows 1 and 3, whilst 𝑈 = 18.92 ms-1 and

18.05 ms-1 considerably exceed the probability of the design wind speed as reported in row 1.

The joint occurrence of earthquake and wind events listed on each column of Table 7 therefore match

the single probability associated to the design wind load, and define a sub-set of multi-hazard scenarios

that tend to magnify the strength and ductility demand of wind turbine towers, as discussed in the

following section.

5. Strength and ductility demands associated to earthquake and wind load

The relationship between strength and ductility determine the performance level of structures. Although

such parameterisation is more common in earthquake than wind applications, both earthquake and wind

effects can be merged to quantify linear-elastic and plastic structural performance through strength

reduction factors, as in Martinez-Vazquez (2017). In the present investigation the strength demands

imposed by multi-hazard conditions was quantified in simple terms through the second moment of area

that is required to ensure fully-operational conditions of wind energy infrastructure. On a second stage,

a map was established between the estimated demands of strength and the corresponding ductility level.

This enabled to assess the performance level of the wind turbine towers subject to earthquake and wind

joint effects.

5.1 Strength Demand

The relationship FEQ*/ FW

* provided in Table 5 was taken to find the required section of prismatic steel

towers so that the ratio between the maximum displacement and the displacement that induces the yield

stress on the base material, match. To that end, wind and earthquake forces were estimated with Eq. (6).

𝐹𝑊∗ =

𝐹𝐸𝑄∗ + 𝐹𝑊

∗

𝐹𝐸𝑄∗

𝐹𝑊∗⁄ + 1

(6a)

𝐹𝐸𝑄∗ =

𝐹𝐸𝑄∗ +𝐹𝑊

∗

𝐹𝑊∗

𝐹𝐸𝑄∗⁄ +1

(6b)

Strength demands across the three wind turbine towers were defined in terms of the ratio

𝐼𝐸𝑄+𝑊 𝐼𝐸𝑄 𝑜𝑟 𝑊⁄ , where 𝐼𝐽 is the second moment of area associated to material yielding, given the load

j. The strength demand associated to design earthquake (𝐼𝐸𝑄) or wind load (𝐼𝑊) is thus magnified when

this relationship exceeds 1. Results of this analysis are shown in Table 8 considering two load scenarios:

average values associated all 40 earthquake records listed in Table 1 (EQ#1-40), and values

characterising isolated events, namely, EQ#5, EQ#24, and EQ#38. All these combinations being

consistent with the probability analysis reflected in Table 7. For example, EQ#5 which has a magnitude

Ms = 5.3, would be exceeded every year in earthquake prone areas (Global Data) as it is lower than Ms

= 6.36, therefore it can be combined with any wind event. EQ#24 with Ms = 6.53 can be combined with

wind events having �̅� ≤ 18.92 ms-1 whereas EQ#38 with Ms = 7.01 can be combined with wind events

having �̅� ≤ 18.05 ms-1. It is also worth noting that values in Table 8 which are higher than 1 indicate

that the multi-hazard condition would exceed the design strength, 𝐼𝐸𝑄 𝑜𝑟 𝑊.

TABLE 8

From Table 8 it becomes apparent that wind turbine towers would primarily be designed to withstand

wind load since all values of 𝐼𝐸𝑄+𝑊 𝐼𝐸𝑄⁄ are above 1 whilst most values of 𝐼𝐸𝑄+𝑊 𝐼𝑊⁄ are below 1. The

sub-set 18.05 ms-1 < �̅� < 20 ms-1 however defines a region of magnified strength demands ranging

between 1 < 𝐼𝐸𝑄+𝑊 𝐼𝑊⁄ < 1.206. Note that EQ#24 in Table 8 could not be combined with �̅� = 20 ms-1

because its magnitude exceeds Ms = 6.36 therefore the combined probability of those events exceeds

P50 = 0.02. With the same criteria two other combinations of EQ#38 were discarded. Keeping with wind

resisting design, the combined action of �̅� = 18.92 ms-1 and the average of the set EQ#1-40 would

match the strength demand imposed by the design wind load, whereas the demand of strength would be

exceeded in 20.6% if the same wind speed did occur simultaneously with an earthquake of Ms = 6.53

(e.g. EQ#24). The latter identifies the worst case scenario as derived from this investigation.

5.2 Ductility Demand

In Martinez-Vazquez (2017), a relationship between strength reduction factors (Rμ) associated to multi-

load scenarios and structural ductility (𝜇) is established. The definition of these parameters is given in

Eq. (7) and (8) – where F is the restoring force required to keep inelastic displacements within the limit

of the ductility factor μ and 𝑢𝑦 is the displacement that limits linear elastic structural performance.

𝑅𝜇 =𝐹(𝜇=1)

𝐹(𝜇=𝜇) (7)

𝜇 =𝑢𝑚𝑎𝑥

𝑢𝑦⁄ (8)

In that study 𝛤 =𝑀∗

√𝐻2+𝑊2+𝐿2 is defined to characterise the density of structures - where M* is the

generalised mass and H-W-L are side dimensions of prismatic buildings. The relevant strength reduction

factors (SRFs) presented in Martinez-Vazquez (2017) are reproduced here in Fig. 1 and Fig. 2, for cases

where �̅� = 15 and 20 ms-1 and when 𝛤 takes values of 1.91, 60.43, and 17.78.

FIGURE 1

FIGURE 2

For circular shapes, let 𝛤 =𝑀∗

√𝐻2+𝐷2 - where D represents the diameter of the steel towers. This results

in 𝛤 = 2.33 Ton/m, 2.94 Ton/m, and 4.17 Ton/m, for wind turbine towers of 150 m, 200 m, and 250 m

height, respectively. It follows that the magnification of strength demands highlighted in Table 8 can

be expressed in terms of ductility demand, as suggested by Eq. (7) and (8). By interpolation of the

curves shown in Fig. 1 and 2, the average ductility demands estimated across the three wind turbine

towers can be established. This is shown in Table 9.

TABLE 9

The results shown in Table 9 show that the combined action of wind and earthquakes can trigger

inelastic performance of wind turbine towers i.e. 𝜇 > 1. It is also seen that both strength and ductility

demands equal 1 when �̅� = 18.92 ms-1 and considering all earthquake records in the database - noting

that this wind speed is lower than the assumed design wind speed �̅�50 = 20 ms-1. Furthermore, when

the expected earthquake event i.e. EQ#1-40 occurs simultaneous to �̅�50 the strength and ductility

demand both exceed a design limited by the yield condition in 12% and 37% respectively. If particular

events such as EQ#5, EQ#24, and EQ#38 are considered, strength demand exceedances oscillate

between 3.2% and 20.6% whilst the associated ductility demands would surpass the yield condition in

amounts ranging from 9% to 62%.

6. Final Remarks

This research identifies a narrow yet existing probability that earthquake and wind effects modify the

performance level of wind energy infrastructure. The estimated strength and ductility demand of wind

turbine towers indicate that under certain conditions these can undergo unforeseen inelastic

performance during extreme events. This is not addressed by current engineering practice which is

based on the assumption that multi-hazard scenarios are extremely rare and that no relationship can be

established between earthquake and wind events. The evidence however demonstrates that the

frequency of occurrence of earthquakes of medium to high intensity is not rare. According to NOAA

(2017) a total of 1142 earthquakes of 5.3 < Ms < 7.01 occurred across the world in the last 50 years

(1966-2016) with as many as 173 recorded in the last 5 years (2011-2016) and 28 of those in 2016. The

situation depicted in this investigation could be worse if we consider earthquake events of higher

magnitude. NOAA (2017) reports a total of 418 ground motions of 7.01 < Ms < 9.9 in the last 50 years

whilst 62 and 11 of those events were seen in the period 2011-2016 and during 2016, respectively. The

recent events recorded in Mexico included two major earthquakes within a period of two weeks in

September 2017 (CNN, 2017) with one of them nearly clashing with hurricane Katia on 8 September

2017 (ABC News, 2017). This adds further arguments against current design assumptions which ignore

multi-hazard scenarios. It seems therefore necessary to consider regional risk to extreme events in the

formulation of a more robust design framework for practical use.

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List of Tables

Table 1: Earthquake record database

Table 2: Theoretical and simulated mean velocity �̅� (ms-1) and turbulence intensities

Table 3: Cross correlation results amongst the points of interest

Table 4: Geometry and natural frequency of wind turbines

Table 5: Total generalized forces (FEQ* + FW

*) acting on wind turbine towers (kN)

Table 6: Dynamic response amplitudes (m) calculated for wind turbine towers subject to FEQ* + FW

*

Table 7: Probabilities associated to earthquake and wind events, and earthquake magnitudes across

countries

Table 8: Strength demands derived from the combined action of earthquake and wind

Table 9: Strength and ductility demands derived from the combined action of earthquake and wind

Table 1. Earthquake record database

# Earthquake(s) Magnitude Epicentral

Distance Km

vs30

ms-1

PGA

g

1-2 Helena Montana-01, 10/31/1935, Carroll College, 180 / 270 6 2.86 / 2.92 593.35/551.82 0.16

3-4 Northwest Calif-01, 9/12/1938, Ferndale City Hall, 45 / 224 5.5 / 5.8 53.88 / 53.77 219.31 0.15 / 0.11

5-6 Izmir Turkey, 12/16/1977, Izmir, L / T 5.3 3.21 535.24 0.42 / 0.13

7-8 Dursunbey Turkey, 7/18/1979, Dursunbey, L / T 5.34 9.15 585.04 0.18 / 0.24

9-10 Imperial Valley-02, 5/19/1940, El Centro Array #9, 180 / 270 6.95 6.09 213.44 0.25 / 0.15

11-12 Northern Calif-01, 10/3/1941, Ferndale City Hall, 225 / 315 6.4 44.68 219.31 0.10 / 0.12

13-14 Northern Calif-03, 12/21/1954, Ferndale City Hall, 44 / 314 6.5 27.02 219.31 0.16

15 Borrego Mtn, 4/9/1968, El Centro Array #9, 180 6.63 45.66 213.44 0.13

16-17 San Fernando, 2/9/1971, Castaic - Old Ridge Route, 21 / 291 6.61 22.63 450.28 0.32 / 0.28

18-19 San Fernando, 2/9/1971, LA - Hollywood Stor FF, 90 / 180 6.61 22.77 316.46 0.22 / 0.16

20 San Fernando, 2/9/1971, Lake Hughes #1, 21 6.61 27.4 425.34 0.15

21-22 San Fernando, 2/9/1971, Lake Hughes #12, 21 / 291 6.61 19.3 602.1 0.38 / 0.28

23-24 Imperial Valley-06, 10/15/1979, Bonds Corner, 140 / 230 6.53 2.66 223.03 0.52 / 0.77

25-26 Imperial Valley-06, 10/15/1979, El Centro Array #4, 140 6.53 7.05 208.91 0.48 / 0.27

27-28 Imperial Valley-06, 10/15/1979, El Centro Array #5, 140 / 230 6.53 3.95 205.63 0.33 / 0.38

29-30 Imperial Valley-06, 10/15/1979, El Centro Array #7, 140 / 230 6.53 0.56 210.51 0.34 / 0.47

31-32 Kern County, 7/21/1952, Taft Lincoln School, 21 / 111 7.36 38.89 385.43 0.14 / 0.15

33-34 Taiwan SMART1(45), 11/14/1986, SMART1 C00, EW / NS 7.3 56.01 309.41 0.12 / 0.15

35-36 Taiwan SMART1(45), 11/14/1986, SMART1 O02, EW / NS 7.3 57.13 285.09 0.16 / 0.24

37-38 Cape Mendocino, 4/25/1992, Petrolia, 0 / 90 7.01 8.18 422.17 0.58 / 0.66

39-40 Landers, 6/28/1992, Lucerne, 260 / 345 7.28 2.19 1369 0.65 / 0.61

Table 2. Theoretical and simulated mean velocity �̅� (ms-1) and turbulence intensities

Stats \ z (m) 10 40 75 100 140 170 200 210 220 240 250

�̅�t 20.00 30.21 36.21 39.79 42.95 45.39 47.77 48.52 49.15 50.18 50.87

�̅�s 19.86 29.88 35.14 38.62 41.69 44.05 46.32 47.07 47.69 48.69 49.35

Iu,t 0.295 0.206 0.244 0.221 0.195 0.172 0.146 0.137 0.130 0.116 0.107

Iu,s 0.295 0.206 0.173 0.153 0.135 0.122 0.108 0.104 0.100 0.093 0.088

Table 3. Cross correlation results amongst the points of interest

1 2 3 4 5 6 7 8 9 10 11

1 1.0000 0.4737 0.4100 0.2255 0.2134 -0.0052 -0.0591 -0.1249 -0.1843 -0.1688 -0.0237

2 0.4237 1.0000 0.6224 0.3814 0.2928 0.1594 0.0892 0.0231 -0.0493 -0.0151 0.0716

3 0.2090 0.4767 1.0000 0.6510 0.5322 0.2694 0.2860 0.2238 0.1078 0.1387 0.2322

4 0.1176 0.2605 0.5419 1.0000 0.7333 0.4118 0.4189 0.3157 0.2204 0.2723 0.2803

5 0.0653 0.1408 0.2895 0.5322 1.0000 0.6122 0.5377 0.4465 0.3681 0.3842 0.3309

6 0.0394 0.0831 0.1690 0.3092 0.5798 1.0000 0.5934 0.5431 0.5346 0.4584 0.3253

7 0.0231 0.0479 0.0961 0.1748 0.3265 0.5625 1.0000 0.8046 0.7388 0.6270 0.5227

8 0.0192 0.0394 0.0788 0.1429 0.2665 0.4588 0.8155 1.0000 0.8113 0.6355 0.5248

9 0.0164 0.0336 0.0669 0.1210 0.2255 0.3879 0.6893 0.8453 1.0000 0.7276 0.5858

10 0.0127 0.0257 0.0508 0.0917 0.1705 0.2929 0.5200 0.6376 0.7543 1.0000 0.7524

11 0.0106 0.0213 0.0420 0.0756 0.1403 0.2408 0.4274 0.5240 0.6198 0.8217 1.0000

Table 4. Geometry and natural frequency of wind turbines

ID # Height

(m) Dbase (m) Dtop (m) tbase (m) ttop (m) n0 (Hz)

1 150 7.5 4.0 0.05 0.016 0.44

2 200 10 7.5 0.075 0.018 0.34

3 250 15 10 0.10 0.025 0.24

Table 5. Total generalized forces (FEQ* + FW

*) acting on wind turbine towers (kN)

𝑈 H = 150 m H = 200 m H = 250 m Average FEQ* / FW

* ms-1 mean rms peak mean rms peak mean rms peak mean rms peak zero 0.11 9.78 92.5 0.29 30.8 284 0.78 70.8 652 - - -

0.5 0.39 9.79 92.6 0.95 26.3 249 1.82 60.2 572 10.8 167 193

2.5 9.08 9.97 96.1 22.1 26.7 256 41.8 60.9 584 0.43 6.68 7.72

5 36.3 12 118 88.3 30.9 304 167 66.9 667 0.11 1.67 1.93

10 145 28 234 353 64.9 570 667 120 1137 0.03 0.42 0.48

15 326 58 471 794 133 1139 1500 235 2150 0.01 0.19 0.21

20 580 101 822 1411 232 1987 2666 405 3712 0.01 0.10 0.12

Table 6. Dynamic response amplitudes (m) calculated for wind turbine towers subject to FEQ* + FW

*

𝑈 H = 150 m H = 200 m H = 250 m ms-1 mean rms peak mean rms peak mean rms peak zero 9.3x10-6 0.003 0.014 2.5 x10-5 0.005 0.017 2.6 x10-5 0.006 0.021

0.5 1.4 x10-4 0.003 0.015 2.2 x10-4 0.005 0.022 3.6 x10-4 0.007 0.026

2.5 0.003 0.004 0.017 0.005 0.006 0.023 0.009 0.009 0.029

5 0.013 0.006 0.030 0.021 0.009 0.046 0.034 0.015 0.067

10 0.051 0.016 0.090 0.083 0.030 0.155 0.137 0.046 0.229

15 0.116 0.035 0.197 0.187 0.067 0.342 0.309 0.101 0.507

20 0.205 0.061 0.349 0.333 0.119 0.605 0.549 0.179 0.898

Table 7. Probabilities associated to earthquake and wind events, and earthquake magnitudes across countries

Probability of Exceedance, Seismic Magnitude per Country, or Wind Velocity a b

PW 0.02 0.05 0.1 0.25 0.5 0.75 0.9 1 - -

Cprob 1 0.95 0.90 0.84 0.78 0.75 0.68 0.52 - -

�̅� (ms-1) 20 18.92 18.05 16.75 15.53 14.49 13.68 10.33 - -

PEQ 1 0.4 0.2 0.08 0.04 0.027 0.022 0.02 - -

India1 3.19 3.67 4.04 4.52 4.88 5.09 5.19 5.24 4.35 0.83

UK2 2.06 2.44 2.74 3.12 3.42 3.59 3.66 3.71 3.82 1.03

Turkey3 1.96 2.48 2.87 3.39 3.78 4.01 4.11 4.17 3.21 0.77

Greece4 4.86 5.13 5.33 5.59 5.79 5.91 5.96 6.00 8.99 1.5

Switzerland5

3.54 3.96 4.27 4.68 5.00 5.18 5.26 5.31 5.1 0.96

Global

Data1 6.36 6.73 7.02 7.39 7.68 7.84 7.92 7.96

8.44 1.06

1 Sharma et al. (1999); 2 NERC (2017); 3 Bayrak et al. (2008); 4 Papazachos et al. (1997); 5 Wiemer (2000)

Table 8. Strength demands derived from the combined action of earthquake and wind

Load Scenario Wind Speed 𝑈 in ms-1

0.00 10.33 13.68 14.49 15.53 16.75 18.05 18.92 20.00

𝐼𝐸𝑄+𝑊 𝐼𝐸𝑄⁄ EQ#1-40 1.0 9.36 16.13 18.07 20.73 24.11 27.98 30.76 34.43

𝐼𝐸𝑄+𝑊 𝐼𝑊⁄

EQ#1-40 - 0.305 0.525 0.588 0.674 0.784 0.910 1.00 1.12

EQ#5

Ms = 5.3 - 0.314 0.541 0.606 0.696 0.809 0.939 1.032 1.157

EQ#24

Ms = 6.53 - 0.367 0.632 0.708 0.813 0.945 1.097 1.206 -

EQ#38

Ms = 7.01 - 0.346 0.596 0.668 0.766 0.890 1.033 - -

Table 9. Strength and ductility demands derived from the combined action of earthquake and wind

Wind Speed 𝑈 in ms-1

18.05 18.92 20.00 Earthquake Event EQ#24 EQ#38 EQ#1-40 EQ#5 EQ#24 EQ#1-40 EQ#5

Strength Demand 𝐼𝐸𝑄+𝑊 𝐼𝐸𝑄⁄

1.097 1.033 1.00 1.032 1.206 1.12 1.157

Ductility Demand

𝜇 = 𝑢𝑚𝑎𝑥 𝑢𝑦⁄ 1.27 1.09 1.00 1.09 1.62 1.37 1.49

List of Figures

Figure 1. Strength reduction factors estimated for when �̅� = 15 𝑚𝑠−1

(Adapted from Martinez-Vazquez, 2017)

Figure 2. Strength reduction factors estimated for when �̅� = 20 𝑚𝑠−1

(Adapted from Martinez-Vazquez, 2017)

(a) Γ = 60.43, 𝑈 = 15 𝑚𝑠−1 (b) Γ = 17.78, 𝑈 = 15 𝑚𝑠−1 (c) Γ = 1.91, 𝑈 = 15 𝑚𝑠−1

Fig. 1. Strength reduction factors estimated for when 𝑈 = 15 𝑚𝑠−1

(Adapted from Martinez-Vazquez, 2017)

(a) Γ = 60.43, 𝑈 = 20 𝑚𝑠−1 (b) Γ = 17.78, 𝑈 = 20 𝑚𝑠−1 (c) Γ = 1.91, �̅� = 20 𝑚𝑠−1

Fig. 2. Strength reduction factors estimated for when 𝑈 = 20 𝑚𝑠−1

(Adapted from Martinez-Vazquez, 2017)

0

1

2

3

4

5

0 1 2 3 4 5

R'm

T (s)

m = 1 m = 2

m = 3 m = 4

m = 5 m = 6

0

1

2

3

4

5

0 1 2 3 4 5

R'm

T (s)

m = 1 m = 2

m = 3 m = 4

m = 5 m = 6

0

1

2

3

4

5

0 1 2 3 4 5

R'm

T (s)

m = 1 m = 2

m = 3 m = 4

m = 5 m = 6

0

1

2

3

4

5

0 1 2 3 4 5

R'm

T (s)

m = 1 m = 2

m = 3 m = 4

m = 5 m = 6

0

1

2

3

4

5

0 1 2 3 4 5

R'm

T (s)

m = 1 m = 2

m = 3 m = 4

m = 5 m = 6

0

1

2

3

4

5

0 1 2 3 4 5

R'm

T (s)

m = 1 m = 2

m = 3 m = 4

m = 5 m = 6

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